Abstract

Quantum mechanical corrections to the hydromagnetic force balance equation, derived from the microscopic Schrödinger-Pauli theory of quantum plasmas, modify the equilibrium structure and hence the mass quadrupole moment of a neutron star. It is shown here that the dominant effect — spin paramagnetism — is most significant in a magnetar, where one typically has μB|B|≳kBTe, where μB is the Bohr magneton, B is the magnetic field, and Te is the electron temperature. The spin paramagnetic deformation of a nonbarotropic magnetar with a linked poloidal-toroidal magnetic field is calculated to be up to ∼10 times greater than the deformation caused solely by the Lorentz force. It depends on the degree of Pauli blocking by conduction electrons and the propensity to form magnetic domains, processes which are incompletely modelled at magnetar field strengths. The star becomes more oblate, as the toroidal field component strengthens. The result implies that existing classical predictions underestimate the maximum strength of the gravitational wave signal from rapidly spinning magnetars at birth. Turning the argument around, future gravitational-wave upper limits of increasing sensitivity will place ever-stricter constraints on the physics of Pauli blocking and magnetic domain formation under magnetar conditions.

MHD equilibria are important to calculate for several reasons. As in this paper, they are a key input into calculations of the star’s mass ellipticity and hence its gravitational wave luminosity (Cutler, 2002; Stella et al., 2005; Dall’Osso, Shore, & Stella, 2009). In addition, they are a starting point for assessing the long-term stability of the magnetic field and predicting energy releases during magnetar bursts. Field stability can be tested by numerical simulations (Braithwaite & Nordlund, 2006; Braithwaite, 2009; Armaza, Reisenegger & Valdivia, 2014) or analytically (Akgün et al., 2013). Magnetar burst observations, in particular the 1998 August 27 flare from SGR 1900+14, offer strong evidence for the existence of strong internal magnetic fields, and the bursts themselves can be interpreted as transitions between internal magnetic states (Ioka, 2001; Corsi & Owen, 2011). MHD equilibria also serve as a starting point for simulations of Hall drift (Marchant et al., 2014). It is timely, therefore, to take into account as many realistic physical effects as possible when modelling neutron stars in hydromagnetic equilibrium.

Previous calculations of the magnetic deformation of a neutron star, including those referenced above, have neglected quantum mechanical corrections to the MHD equations of motion (with the important exception of superconductivity). In many applications, this is entirely justified. In most ordinary neutron stars, for example, the surface dipole magnetic field strength does not exceed the critical value Bc=m2ec2/eℏ=4.4×109 T, where quantum electrodynamic processes are activated (Melrose, 2010). Unless the internal field is much stronger than the observed dipole, spin-related effects are washed out by thermal fluctuations in these objects. Moreover many-body quantum correlations, which can lead to macroscopic effects in principle, are typically nullified in the hydrodynamic regime in practice (again, with the important exception of superconductivity). In magnetars, however, Bc is comfortably exceeded, and the thermal energy per dynamical degree of freedom is less than the Larmor energy. Quantum kinetic effects, especially those involving spin, become important under such conditions and modify the ‘macroscopic’ MHD equations (de Groot & Suttorp, 1972; Brodin & Marklund, 2007b). The hydrodynamic description of quantum plasmas, including macroscopic spin effects (Haas, Manfredi, & Feix 2000; Brodin & Marklund 2007, hereafter BM07; Marklund & Brodin 2007), enjoys a variety of applications to multi-stream instabilities (Anderson et al., 2002), linear and nonlinear ion-acoustic waves (Haas et al., 2003), four-wave interactions and nonlinear Zakharov wave collapse (Garcia et al., 2005), quantum dusty plasmas (Shukla & Stenflo, 2006), laser-plasma experiments (Mourou, Tajima, & Bulanov, 2006), and fusion plasmas (Cowley, Kulsrud, & Valeo, 1986).

In this paper, we calculate how one particular quantum effect — spin paramagnetism — modifies the MHD equilibrium structure of a magnetar. We focus on spin paramagnetism, because it is the leading quantum force correction relevant to magnetars within the framework of spin MHD (BM07). It sets the stage for future analyses of other quantum corrections, which lie outside the scope of this paper. In Sec. 2.1 and Appendix A, we review briefly the hydrodynamic description of quantum plasmas and its self-consistent derivation from the underlying, microscopic, quantum theory. In Sections 2.2 and 2.3, we estimate the order of magnitude of the spin paramagnetic corrections in a magnetar and some possible saturation mechanisms. In Sec. 3, we construct, as a worked example, a modified MHD equilibrium for a nonbarotropic star with a linked poloidal-toroidal magnetic field akin to those seen in simulations (Sec. 3.1) and calculate its ellipticity (Sec. 3.2). The results are compared with previous classical calculations in Sec. 4. The consequences for gravitational radiation are discussed briefly. Throughout this work we employ SI units.

2.1 Spin paramagnetism

The multi-fluid equations of motion for a quantum plasma can be derived systematically from the N-particle Schrödinger-Pauli equation including spin by following the prescription in BM07. The main steps in the derivation are summarised in Appendix A. In brief, for each species, one applies a Madelung transformation to the Schrödinger-Pauli equation to write the amplitude of the N-particle wavefunction in terms of number density and the gradient of the phase in terms of bulk velocity. The number density and bulk velocity are averaged over the N-particle ensemble, with each term weighted by the corresponding single-particle probability. The N-particle wavefunction factorizes, if entanglement is negligible (i.e., if the decoherence time is short). The resulting equations of motion are given by equations (A6)–(A8) in Appendix A for each species. The terms on the right-hand sides are too complicated to write out in full; their forms are given explicitly by BM07. Physically, as well as the standard classical pressure and electromagnetic forces, there are macroscopic quantum mechanical forces arising from spin-spin self-correlations and correlations between spins and thermal velocity fluctuations. Related spin-spin and spin-thermal torques are exerted on the ensemble-averaged spin vector.

If the plasma is quasineutral, equations (A6)–(A8) for the protons and electrons can be combined into a single-fluid description like in classical MHD. Letting ρ, p, and v be the mass density, pressure, and bulk velocity of the single MHD fluid respectively, the momentum equation reads

ρ(∂∂t+v⋅∇)v=J×B−ρ∇Φ−∇P+FQ.

(1)

In a neutron star, the forces in (1) are felt specifically by the MHD proton-electron fluid, which typically amounts to a few per cent of the star’s mass. In what follows, therefore, we approximate ρ by the proton mass density ρp when applying equation (1). The first three terms on the right-hand side of (1) are classical, representing the Lorentz force, the gravitational force (Newtonian potential Φ), and pressure gradient respectively (the anisotropic pressure tensor is dropped for simplicity).

The last term on the right-hand side of equation (1) is the quantum force density FQ. In general, it takes a complicated form [see equation (22) of BM07] involving spin-spin interactions and spin-thermal coupling. The latter effect enters implicitly through the spin transport equation (A11). However, in the MHD limit, where the Larmor radius is small compared to the length-scale of magnetic gradients, one can neglect the spin-spin, spin-thermal, and spin inertia terms to a good approximation (see Appendix A and BM07), and the quantum force density acting on the MHD proton-electron fluid (cf. Sec. 3) reduces to

FQ=ρpmp⎡⎣∇⎛⎝ℏ22mpρ1/2p∇2ρ1/2p⎞⎠+G(|B|,Te)μB∇|B|⎤⎦,

(2)

where m denotes the particle mass, μB=eℏ/(2me) is the Bohr magneton, T is the temperature, G(|B|,Te) is the Brillouin function, the subscripts p and e refer to the proton and electron components respectively, and ρp(≫ρe) is the proton mass density. We assume that the particles are non-relativistic, like BM07, although this may not be true closer to the core of a neutron star. The Brillouin function is a thermodynamic factor that equals the ratio (n0+−n0−)/(n0++n0−), where n0+(n0−) are the number densities of particles in the higher (lower) spin states. The Brillouin function is given in full generality by equation (60) in Zamanian, Marklund, & Brodin (2010) and contains the effects of Landau quantization, spin splitting, and Fermi-Dirac statistics. We consider the limits of this expression in Sections 2.2 and 2.3.

The first term in equation (2) is often referred to as the quantum pressure. It arises physically from the self-attraction between bosons. It dominates at small length-scales and is important in the study of superfluid vortex structure (Donnelly, 1991). The second term represents a collective form of spin paramagnetism. In the regime μB|B|≳kBTe, the electron and proton spins tend to align with B, with the electrons dominating the net polarization (μB≫μp). The spin-polarized proton-electron fluid therefore feels a paramagnetic force in the presence of a magnetic gradient.

It is possible that inside neutron stars the quantum effects discussed here in the presence of strong magnetic fields also influence gravitational interactions. A relativistic and self-consistent analysis would then require a modification to the gravitational potential ϕ leading to additional expressions appearing in the quantum force FQ(Drummond & Hathrell, 1980).

2.2 Suppression of the paramagnetic force and magnetization

The spin-MHD theory, developed by BM07, which is applicable when the length-scales of magnetic gradients exceed the Larmor radius (as in a neutron star), leads to a total magnetization given by equation (43) of BM07:

M=μBρpmpG(|B|,Te)^B.

(3)

In a neutron star, the electrons form a degenerate gas. Pauli suppression is expected to lower |M| by a factor of Te/TFe, where TFe is the Fermi temperature of the electrons [TFe∼1012(ρp/1015 kgm−3)2/3 K], by analogy with the Pauli spin magnetization of the conduction electrons in a metal (Kittel, 1966), although the conduction band is modified strongly by the magnetic field at magnetar field strengths, with uncertain implications for M. Thus, in the regime Te≪TFe (relevant to this paper), the Brillouin function G(|B|,Te), given by equation (60) of Zamanian, Marklund, & Brodin (2010), simplifies into

G(|B|,Te)=32(μB|B|kBTFe),

(4)

which accounts for Pauli blocking. In contrast, in the high-temperature regime Te≫TFe, G(|B|,Te) simplifies into

G(|B|,Te)=tanh(μB|B|kBTe).

(5)

Additionally, for the simplified forms of G(|B|,TFe) given by equations (4) and (5) to be valid, one must have μB|B|<kBTFe, i.e., |B|≲1.5×1012(ρp/1015 kg m−3)2/3 T, which is readily satisfied in most neutron stars. Equations (3)–(5) imply a magnetization-to-magnetic-field ratio μ0|M|/|B|≈7(ρp/1015 kg %
m−3)(TFe/1012 K)−1 for Te≪TFe. Given M, we can calculate the magnetic susceptibility χ=|M|/|H|, where H is the magnetic induction, H=B/μ0−M. In the neutron star regime, where the conditions Te≪TFe and μB|B|<kBTFe are satisfied, we find χ≈2×10−2.

In reality, the Brillouin model underpinning equation (3) may break down under neutron star conditions. The tendency towards alignment is reduced at high densities by the chemical potential, which is subtracted from μB|B| in the Boltzmann probability leading to equation (3). There are also subtle collective effects to consider. Equation (3) does not exhibit the expected oscillation of M versus |B|, known as the de Haas-van Alphen oscillation, caused by changes in the number of occupied Landau levels (Blandford & Hernquist, 1982; Ferrer et al., 2010; Noronha & Shovkovy, 2007; Chatterjee et al., 2015). On the other hand, including the Heisenberg nearest-neighbour exchange interaction in the Brillouin theory tends to enhance spin alignment and encourage magnetic domains to form, a potentially strong effect which is nevertheless hard to quantify (Dong, Zuo, & Gu, 2013). In this way, a more realistic expression for M includes a prefactor K of the form (Kittel, 1966)

M=KμBρpmpG(|B|,Te)^B,

(6)

where K is a parameter which describes additional factors that are potentially absent from equation (60) of Zamanian, Marklund, & Brodin (2010). Gravitational wave limits could be used to constrain the parameter K, since the deformation induced by the paramagnetic force will be directly proportional to the magnetisation (6).

We present some examples of theoretical values of χ(Noronha & Shovkovy, 2007; Skobelev, 2012; Dong, Zuo, & Gu, 2013; Endrődi, 2014; Rabhi et al., 2014) in Table 1 for comparison. As evident, there are disagreements regarding χ and the method of calculating it. We see also that our value for susceptibility, χ∼10−2, in the μB|B|/kBTe≳1 regime is not very different from the values calculated using the methods summarized in Table 1.

All this suggests that the form of FQ given by equation (44) of BM07 and equation (2) in this paper is not the whole story (especially for the low-|B|/high-T regime), but it is hard to be confident about the form and order-of-magnitude of the corrections in the absence of experimental guidance, when the material and physical conditions are so exotic. A self-consistent analysis of spin paramagnetism in a dense, highly-magnetized, multi-species fluid, as in a realistic neutron star, including collective effects and domain formation, lies beyond the scope of this paper. Our aim here is to point out that this force contributes to the stellar ellipticity and to estimate the maximum size of this contribution. The reader should bear in mind that the effect may be suppressed (or enhanced, in the case of domain formation) by the mechanisms in this paragraph and potentially others not referenced here. Some mathematically consistent remedies which may lead to a different expression for the net magnetisation M, and hence an alternate χ for the low-|B|/high-T regime, are explored in the Appendix.

2.3 Order-of-magnitude estimates

Before constructing an MHD equilibrium explicitly as an example, we compare the characteristic magnitudes of the two quantum corrections in equation (2) with the perturbing Lorentz force considered in previous classical analyses of neutron star deformations.

The ratio κ1 of the quantum pressure to the Lorentz force is

κ1

∼

μ0ℏ2ρpmpmeR2B2

(7)

∼

9×10−33(ρp1015 kg m−3)(R104 m)−2(|B|1011 T)−2,

(8)

up to factors of order unity, where we approximate ∇ with R−1, where R is the stellar radius. As expected, κ1 is tiny. The quantum pressure is only important in vortex cores with diameter ≲10−11 m.

The ratio κ2 of the paramagnetic force to the Lorentz force is

κ2

∼

32μ2Bμ0ρpkBmpTFe

(9)

∼

7(ρp1015 kg m−3)(TFe1012 K)−1,

(10)

up to factors of order unity. The estimates in equations (9) and (10) is made for the regime Te≪TFe, where G(|B|,Te)=(3/2)(μB|B|/kBTFe)(Zamanian, Marklund, & Brodin, 2010).

Equation (10) demonstrates three important points. First, for typical magnetar fields, the spin paramagnetic correction to the Lorentz force J×B is large (Brodin & Marklund, 2007b). The ratio |FQ|/|J×B| corresponds to the ratio of the electron Larmor energy to the thermal energy per degree of freedom, modified by Pauli blocking (Section 2.2). Second, the correction is appreciable even with respect to the ‘background’ nonmagnetic forces. For example, the ratio of the spin paramagnetic force to the pressure gradient ∇P∼∇(ρkBTe/mp) evaluates approximately to μBB/(kBTe), which can exceed unity in a magnetar. Third, the correction nominally remains relevant in ordinary neutron stars (B≲108 T), since κ2 only depends on ρp and TFe. This counterintuitive result essentially arises because neutron star matter is so dense that even the weak residual alignment of spins in the regime μB|B|≲kBTe is enough to produce a significant magnetization per unit volume. In any event, neither spin paramagnetism nor the Lorentz force produce astrophysically interesting ellipticities, ≳10−6 (that is to say, interesting from the gravitational wave viewpoint), for |B|≲107 T.

3 Worked example: nonbarotropic star with a linked poloidal-toroidal magnetic field

In this section, we calculate the ellipticity produced by the quantum force density FQ for a linked poloidal-toroidal field of the form considered by many authors (Braithwaite & Nordlund, 2006; Braithwaite, 2009; Mastrano et al., 2011; Dall’Osso et al., 2015). The single-fluid form of FQ in equation (2) acts on the electron-proton MHD fluid. The neutron condensate is also deformed by the spin paramagnetic force, but the form of FQ is different and more complicated [see equation (22) of BM07]. Luckily, the effect on the neutrons is of order (nn/ne)(μn/μB)∼10−2 times the effect on the protons, where nn (ne) and μn (μB) are the number density and magnetic moment of the neutrons (electrons) respectively. It is therefore negligible in magnetar ellipticity calculations.

where M⋆ is the stellar mass and R is the radius. We close the system with the Poisson equation ∇2Φ=4πGρ. This background ‘parabolic’ density profile is chosen for analytic simplicity, but Mastrano et al. (2011) showed a posteriori that the induced ellipticity is within 5 per cent of that obtained for a more realistic, polytropic density profile.

Introducing the magnetic field as the source of the perturbation, we write ρ↦ρ+δρ along with P↦P+δP. We take a poloidal and toroidal decomposition of an axisymmetric magnetic field (Chandrasekhar, 1956),

B(r,θ)=B0[ηp∇α×∇ϕ+ηtβ(α)∇ϕ],

(13)

where the flux function α(r,θ) sets the poloidal field structure, and the dimensionless parameters ηp and ηt define the relative magnitudes of the poloidal and toroidal components respectively. The function β, which defines the toroidal field, must be a function of α to ensure that the azimuthal component of the Lorentz force vanishes, since there is no other azimuthal force to balance it in the axisymmetric MHD equilibrium. We further assume a dipole magnetic field, for which we may take (Mastrano et al., 2011)

the toroidal component of B is confined to a toroidal region (α⩾1) inside the star around the neutral curve;

J=μ−10∇×B is finite and continuous everywhere inside the star and vanishes at the surface (r=1).

Keeping terms linear in the density and pressure while employing the Cowling approximation (δΦ=0), the force balance equation (1) now reads

1μ0(∇×B)×B+μBρpmpG(|B|,Te)∇|B|=∇δP+δρ∇Φ,

(16)

where we neglect the quantum pressure term (κ1≪κ2). We solve for δρ by taking the curl of both sides of equation (16), matching the ϕ-components, and then integrating with respect to θ(Mastrano et al., 2011; Mastrano, Lasky, & Melatos, 2013). In general, for any deforming axisymmetric net force F on the left-hand side of (16), we have

∂δρ∂θ=−rRdrdΦ(∇×F)ϕ.

(17)

Figure 1: Logarithmic plot of the absolute value of mass ellipticity ϵ versus Λ, the ratio of poloidal to total magnetic field energy, for canonical magnetar parameters B0=5×1010 T, R=104 m, M=1.4M⊙ and Te=107 K. The thin solid curve corresponds to |ϵB|, the deformation due solely to the Lorentz force exerted by a dipolar poloidal-toroidal field (Mastrano et al., 2011), for ϵB>0. The thin dotted curve corresponds to |ϵB| for ϵB<0. The thin dashed curve corresponds to |ϵQ|, the deformation due solely to spin paramagnetism, for ϵQ>0. The thin dashed-dotted curve corresponds to |ϵQ| for ϵQ<0. ϵQ is directly proportional to np/nn; we use np/nn=10−3 here. The thick solid curve shows the total ellipticity |ϵ|=|ϵB+ϵQ|. Note that ϵ is always positive, but ϵB and ϵQ can be positive or negative.

3.2 Ellipticity

The stellar deformation is characterized by the mass ellipticity (Abbott et al., 2008)

ϵ=Izz−IxxI0,

(18)

where I0 is the moment of inertia of the unperturbed spherical star, and the moment-of-inertia tensor is given by

Ijk=R5∫Vd3x[ρ(r)+δρ(r,θ)](r2δjk−xjxk),

(19)

where the integral is taken over the volume of the star (r⩽1). We calculate ϵ by solving (16) for δρ and substituting into (19). One picks up an integration constant when solving (16), which is needed to ensure that δρ is continuous across α=1. If the magnetic field contains higher-order multipoles, it is not always possible to find an integration constant that ensures continuity of δρ, except in some special cases (Mastrano, Suvorov, & Melatos, 2014). Note that I0 equals nnmpR5 up to a multiplicative factor of order unity (nn≫np), whereas Izz−Ixx is proportional to npmpR5 for the electron-proton MHD fluid described by equations (2) and (16). Hence ϵ scales as np/nn overall. The neutrons are deformed by FQ too, but this effect is small, as discussed in the first paragraph of Sec. 3.

In Fig. 1, we plot ϵQ for np/nn=10−3 and ϵB versus Λ=Ep/(Ep+Et), where ϵQ is the ellipticity due solely to the spin paramagnetic effect, ϵB is the ellipticity due solely to a magnetic dipole field, and Ep and Et are the total magnetic energy densities of the poloidal and toroidal components respectively. We plot ϵQ and ϵB separately to draw attention to their different behaviours; we also plot the the total ellipticity ϵ, the sum of ϵQ and ϵB, in the same figure. The Cowling approximation can change the value of ϵB by a factor of ∼2 for the strongest magnetar fields (Yoshida, 2013). We take the mass, radius, temperature, and surface equatorial magnetic field strength to be 1.4M⊙, 104 m, 107 K, and 5×1010 T respectively. In the regime Te≪TFe, e.g. in a magnetar, ϵQ is independent of Te. The deformation ϵQ scales as np/nn if the deformation of the neutron fluid is neglected (see Sec. 3). For the canonical magnetar case shown in Fig. 1, we derive

ϵQ=−1.12×10−2(npnn)(B05×1010 T)2(M∗1.4M⊙)−1(R104 m)(1−0.83Λ).

(20)

We see that FQ alone tends to deform the star into a prolate shape for Λ≳0.83 and into an oblate shape for Λ≲0.83. This is the opposite behaviour of the Lorentz force. We also see that, in general, the maximum spin paramagnetic deformation (in the absence of Pauli blocking) is greater than that caused by the Lorentz force (|ϵQ|>|ϵB|) for Λ≲0.68. Note however that equilibria with Λ≲0.01 are not expected to be stable (Braithwaite, 2009; Akgün et al., 2013).

Equation (19) describes the mass-density contribution to the moment of inertia arising from the ρc2 term in the T00 component of the stress-energy tensor (Thorne, 1980). In reality, there is also a direct electromagnetic contribution to Ijk and hence ϵ arising from the B2/2μ0 term in T00; see equation (148) in Chapter 5 of de Groot & Suttorp (1972). It can be shown, however, that this contribution only changes ϵ by about ∼2 per cent (Mastrano, Suvorov, & Melatos, 2014). Note that Tμν contains terms involving S in general but not in the T00 component measured in the bulk frame.

In this paper, we calculate the ellipticity of a strongly magnetized neutron star taking into account spin paramagnetism, the leading quantum mechanical correction to the MHD equations of motion. The maximum size of this correction exceeds the classical Lorentz force (J×B), when the condition μB|B|>kBTe is satisfied, as routinely occurs in a typical magnetar. The correction arises physically because the electron spins (and, to a lesser extent, the proton spins) polarize the plasma, which then feels a force in a magnetic gradient. Whether or not the correction reaches its maximum under realistic, astrophysical conditions depends on poorly understood physics like magnetic domain formation in a magnetar, which we do not attempt to model here.

As a worked example, we calculate the maximum ellipticity ϵQ due to FQ versus the poloidal-to-total magnetic energy ratio Λ for a canonical magnetar. The behaviour of ϵQ as a function of Λ is markedly different from the classical case. For example, the star becomes more oblate as the toroidal field strength increases (i.e., as Λ decreases), unlike in the classical case (shown as the thin solid and thin dotted curves in Fig. 1). For np/nn=10−3, the star becomes oblate under FQ alone for Λ≳0.83. For most values of Λ, FQ has a stronger effect on ϵ than the Lorentz force. We find typically ϵQ≈−1.12×10−2(np/nn)(B0/5×1010 T)2 in the magnetar regime. In the Brillouin approximation used in this paper, the magnetization is independent of Te in the regime Te≪TFe. Specifically, therefore, the surface temperature of magnetars should not alter the deformation substantially. In weaker field stars, with μB|B|≲kBTe, it may play a larger role. We find that we need (ρp/1015 kg m3)(TFe/1012 K)−1≳0.14 to obtain |FQ|>|J×B|. We remind the reader that the foregoing values of |FQ| are maxima; in reality |FQ| depends on other collective effects (some of which are discussed in Sec. 2.2), which are not completely modelled at magnetar field strengths.

Not only does FQ increase |ϵ| by about one order of magnitude relative to J×B (Fig. 1), it also changes the shape of the star. The Lorentz force alone yields a prolate star for Λ≲0.38 and an oblate star for Λ≳0.38. Adding FQ leads to an oblate star for all Λ. Cutler (2002) predicted that the wobble angle of a precessing prolate star with misaligned angular momentum and magnetic axes tends to grow, until these axes are orthogonal, which is the optimal state for gravitational wave emission. Thus, FQ, even as it increases |ϵ|, may make detection of gravitational waves from magnetar-like sources more difficult.8

Although it is not known with certainty that the protons in the interiors of neutron stars (and in particular magnetars) form a type II superconductor, there is circumstantial evidence for the thesis from X-ray measurements of the cooling rate of the central compact object in the supernova remnant Cassiopeia A (Yakovlev, Levenfish, & Shibanov, 1999; Ho & Heinke, 2009). It is therefore worth comparing the spin paramagnetic force density with the standard superconducting terms like μ−10B⋅∇(Hc1^B), which exceed J×B by a factor Hc1/B. From equation (95) of Glampedakis, Andersson, & Samuelsson (2011), we find |FQ|/|μ−10B⋅∇(Hc1^B)|∼(3/2)μ0μ2Bρp|B|/(mpkBTFeHc1)∼7×102(ρp/1017 kg m−3)(TFe/1012 K)−1(|B|/Hc1). Hence the spin paramagnetic correction is comparable in magnitude to the modified Lorentz force in a superconductor for typical magnetar parameters with Hc1∼B and ρp∼10−3ρ.

Because the signal-to-noise ratio S/N of a gravitational wave source is directly proportional to |ϵ|, we expect the spin paramagnetic force to enhance the S/N of magnetars significantly (especially older, cooler ones), even for ranges of Λ where ϵB is expected to be small. For example, if np/nn=10−3, for Λ=0.4, we find |ϵB|=1.6×10−7 but |ϵQ|=1.2×10−5; for Λ=0.3, we find |ϵB|=1.8×10−6 but |ϵQ|=2×10−5. The most likely magnetar gravitational wave source is a hot, newborn one, hypothesized to spin with an initial period ∼1 ms (Thompson & Duncan, 1993; Dall’Osso, Shore, & Stella, 2009). As a quick example, consider a newborn magnetar in the Virgo cluster, rotating with initial spin period 0.97 ms and final spin period 10 s, with B0=5×1010 T (Dall’Osso, Shore, & Stella, 2009; Mastrano et al., 2011). This magnetar has significant detectability (S/N>10) for Λ≲10−2, which is the lower limit for stability (Braithwaite, 2009; Akgün et al., 2013). With FQ in effect, however, one obtains S/N>10 with Λ≲5×10−2 and np/nn=10−3. Note that ϵQ is independent of temperature, as long as Te≪TFe [see equation (A14) and Zamanian, Marklund, & Brodin (2010)]; TFe is indeed higher than the estimated birth temperature (Te∼1011 K) of neutron stars (Yakovlev & Pethick, 2004; Yakovlev et al., 2004; Dall’Osso, Shore, & Stella, 2009). Gravitational-wave experiments of the kind above may play a role in constraining the uncertain Pauli suppression and domain formation physics described in Sec. 2.3 and Table 1 and may ultimately constitute the main application of our results. The limits implied by gravitational wave experiments with current detectors are above the χ values in Table 1, consistent with at least partial Pauli suppression, but stricter and more interesting limits will follow as gravitational wave detector sensitivities improve.

The self-consistent hydrodynamic theory of a quantum plasma predicts the existence of several other quantum corrections arising from spin-spin and spin-thermal correlations, and from torques on the ensemble-averaged spin vector in the presence of a magnetic gradient. A cursory introduction to these effects is given in Appendix A, together with some key references, but their analysis lies outside the scope of this paper. There may also be nontrivial saturation physics which modifies the Brillouin model of the magnetization, as discussed in Sections 2.2 and 2.3. Our goal here is to alert the reader to the potential importance of quantum corrections and calculate a worked example for one leading effect, namely spin paramagnetism. As discussed in Sections 2.2 and 2.3 and references therein, there are many theoretical and experimental uncertainties regarding the behaviour of matter in the high density, high magnetization regime. The simple Brillouin model used here simply gives the maximum magnetization one can expect. In future, it will be interesting to extend the spin paramagnetic calculation to study different magnetic configuration, higher-order multipoles, and stability, as well as include some of the other quantum corrections discussed in BM07 and collective processes like the formation of magnetic domains.

Acknowledgments

We thank Don Melrose for introducing us to key references in the quantum plasma literature, notably including BM07. We also thank Bryn Haskell, the first reviewer Taner Akgün, and the anonymous second reviewer for their insightful comments, which have significantly improved the quality and clarity of this paper. This work was supported by an Australian Research Council Discovery Project Grant (DP110103347) and an Australian Postgraduate Award.

Appendix A Spin magnetohydrodynamics

In this Appendix, we outline briefly the main steps involved in deriving the hydrodynamic quantum force density FQ in equation (1) from the microscopic Schrödinger-Pauli theory of a quantum plasma. The reader is referred to BM07 and references therein for a full treatment9.

Consider an ensemble of N nonrelativistic, spin-12 particles with mass m, magnetic moment μ, and charge q, labelled by the index α. Neglecting entanglement, as appropriate for a bulk fluid whose decoherence time is short, we can factorise the total system wavefunction Ψ according to Ψ=∏Nα=1Ψ(α), where the single-particle wavefunctions Ψ(α) satisfy, as per equation (9) of BM07,

iℏ∂Ψ(α)∂t=[−ℏ22m(∇−iqℏA)2−μB⋅σ+qϕ]Ψ(α).

(21)

In equation (21), ϕ is the electric scalar potential, A is the magnetic vector potential (with E=−∇ϕ−∂A/∂t and B=∇×A), and σ=(σ1,σ2,σ3) is a vector of Pauli matrices, with Cartesian components

σx=(0110),σy=(0−ii0),σz=(100−1).

(22)

To convert equation (21) into hydrodynamic form, we make a Madelung transformation

Ψ(α)=n1/2(α)exp[iS(α)/ℏ]φ(α),

(23)

where n(α) is the number density of the α-th particle, S(α) is the phase,

is the spin density vector. Upon substituting (23) into (21), the Madelung transformation leads to seven coupled equations of motion representing conservation of particle number, conservation of momentum, and spin transport, given by equations (11), (12), and (17) respectively in BM07. In the hydrodynamic limits, these equations are averaged over the N-particle ensemble according to the prescription in section 2 of BM07, weighting terms in the average by the corresponding single-particle probability. The resulting equations of motion for the ensemble-averaged fields n=⟨n(α)⟩, v=⟨v(α)⟩, and S=⟨S(α)⟩ can be written as

∂n∂t+∇⋅(nv)=0,

(26)

mn(∂∂t+v⋅∇)v=qn(E+v×B)−∇⋅Π−∇P+FQ,

(27)

and

n(∂∂t+v⋅∇)S=−2μnℏB×S−∇⋅K+Ωs,

(28)

In (26)–(28), Π is the trace-free anisotropic pressure tensor, P is the isotropic scalar pressure, FQ is the quantum force density (discussed further below), K is the thermal-spin coupling tensor (ensemble averaged product of thermal velocity and spin perturbations), and Ωs is the nonlinear spin correction (ensemble averaged product of spin perturbations); see section 2 of BM07 for explicit definitions. If the fluid comprises multiple spin-12 species, the equations of motion take the form (26)–(28) for each species, with inter-species collision terms added to equation (27).

In a quasineutral electron-proton plasma (ne≈np), equations (26)–(28) for the electron and proton fluids can be combined into a single-fluid description in the MHD limit, just like in a classical plasma but with spin transport added. The resulting single-fluid MHD equations of motion are

∂ρ∂t+∇⋅(ρv)=0,

(29)

ρ(∂∂t+v⋅∇)v=J×B−∇⋅Π−∇P+FQ,

(30)

and

ρ(∂∂t+v⋅∇)S=meeJ⋅∇S−2μρℏB×S−2m∇⋅K+2mΩs.

(31)

In equations (29)–(31), ρ=mene+mpnp≈mpnp denotes the total mass density, v=(meneve+mpnpvp)/ρ denotes the centre-of-mass velocity, J=−eneve+enpvp denotes the total current density, subscripts e and p label the electron and proton species respectively, Π=Πe+Πp≈Πe and P=Pe+Pp≈Pe are total pressure variables, K≈Ke and Ωs≈Ωse are dominated by thermal-spin and spin-spin coupling within the lighter species, and FQ, the total quantum force density, is discussed further below. We neglect the anisiotropic pressure term ∇⋅Π in this paper for simplicity.

Inside a neutron star, the length-scale of global magnetic gradients is much greater than the classical Larmor radius of the charges, so terms that are quadratic in S can be neglected in the expression for FQ and also in equation (31). Likewise, the spin-thermal coupling is small under these conditions. The spin inertia terms on the left-hand side of (31) are negligible, when the natural time-scale of the problem is much longer than the electron cyclotron frequency, e.g. for steady-state configurations. The spin transport equation then implies S×B=0 and hence

M=ρpμBℏmpS.

(32)

In the standard theory of paramagnetism, one has

S=−ℏ2G(|B|,Te)^B,

(33)

where G(|B|,Te) is the Brillouin function, μB is the electron magnetic moment (Bohr magneton), and Te is the electron temperature. The Brillouin function G(|B|,Te) is the ratio (n0+−n0−)/(n0++n0−) and is given by equation (60) of Zamanian, Marklund, & Brodin (2010). The form of G(|B|,Te) simplifies, depending on the temperature regime (Zamanian, Marklund, & Brodin, 2010), into

{numcases}

G(—B—, T_e)=
32(μB—B—kBTFe)&for Te≪TFe,
tanh(μB—B—kBTe)&for Te≫TFe.
The result in the regime Te≪TFe (which is most relevant to this paper) accounts for Pauli blocking. For the simplified form G=(3/2)(μB|B|/kBTFe) to be valid, one must also have μB|B|<kBTFe, i.e., |B|≲1.5×1012 T. This latter condition is readily satisfied in most neutron stars.

The general expression for FQ, which is the sum of equation (22) in BM07 for electrons and protons and contains complicated spin averages and spin-spin correlations, is too lengthy to write down here. However, it simplifies dramatically in the weak-gradient MHD limit above, where it takes the form

FQ=ρ[∇(ℏ22m2pρ1/2∇2ρ1/2)+G(μB|B|kBTe)μBmp∇|B|].

(34)

To a good approximation, FQ for the electron-proton plasma is dominated by the quantum force on the lighter electrons, given by the two terms in (34). Neutrons also feel a quantum force density given by equation (22) of BM07. This effect is smaller (even though the neutrons are more abundant), as discussed in section 3. We emphasise that ρ in equation (A13) refers only to the charged component of the fluid, not the neutrons. It is therefore replaced by ρp in equations (2) and (15) in the main body of the paper.